A Newton-type midpoint method with high efficiency index

In this paper, we present the local convergence analysis of Werner-King's method to approximate the solution of a nonlinear equation in Banach spaces. We establish the local convergence theorem under conditions on the first and second Fréchet derivatives of the operator involved. The convergence analysis is not based on the Taylor expansions as in the earlier studies (which require the assumptions on the third order Fréchet derivative of the operator involved). Thus our analysis extends the applicability of Werner-King's method. We illustrate our results with numerical examples. Moreover, the dynamics and the basins of attraction are developed and demonstrated.

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“…Proof. The proof is analogous to the proof of Theorem 3 in [11]. But for the completeness we restate the proof.…”

Section: Convergence Analysis Of (3)mentioning

confidence: 79%

On the Order of Convergence and the Dynamics of Werner-King's Method

et al. 2023

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“…They have developed different iterative methods and will continue their work to develop new efficient iterative methods. Recently, an efficient iterative method [9] was presented for the nonlinear equation, which is given by: ) ( ), ( ) ( )…”

Section: Introductionmentioning

confidence: 99%

On an Efficient Iterative Method for Fixed Points

Mohan,

Kumar,

Roy

et al. 2023

Contemp. Math.

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Real-world applications depend heavily on the fixed-point solution. In this paper, we have suggested an effective iterative method for fixed points. We have first given the approximate order of convergence for this method using Taylor’s series. The radii of convergence balls for this method can then be calculated using a local convergence theorem that we then present. The semilocal convergence theorem, which determines the starting point’s accuracy, is then presented. We have created some technical lemmas and theorems to serve this purpose. In contrast to an earlier study using the same type of method for nonlinear equations, we have not used the convergence conditions on higher-order Frechet derivatives in our study of convergence. Finally, some numerical examples are provided to support the theoretical findings we made. This highlights the uniqueness of this study.

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